acuk Received April 22 2008 accepted May 26 2008 posted June 10 2008 Doc ID 95342 published July 11 2008 We extend the design of radially symmetric invisibility cloaks through transformation optics as proposed by Pendry et al Science 312 1780 2006 t ID: 67821
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ElectromagneticanalysisofcylindricalcloaksofanarbitrarycrosssectionAndréNicolet,FrédéricZolla,andSébastienGuenneauInstitutFresnel,UMRCNRS6133,UniversityofAix-Marseille,Case162,F13397MarseilleCedex20,FranceDepartmentofMathematicalSciences,UniversityofLiverpool,PeachStreet,LiverpoolL693BX,UKCorrespondingauthor:guenneau@liverpool.ac.ukReceivedApril22,2008;acceptedMay26,2008;postedJune10,2008(Doc.ID95342);publishedJuly11,2008Weextendthedesignofradiallysymmetricinvisibilitycloaksthroughtransformationopticsasproposedbyetal.al.312,1780(2006)]tocoatedcylindersofanarbitrarycrosssection.Thevalidityofour ellipticalcrosssectionparameterizedas .Wecheckedthatitpro-videsexactlythesamewavetrajectoriesandÞeldproÞleasin[],wheresimilarresultswereobtainedthroughaspacedilatationofacircularcylindricalcloak.Wefurtherretrievedthewavepatternof[whichconsideredaneccentricellipticannularcloak.Toobtaingeneralshapes,aÞniteFourierexpan-maybeused.Forthesakeofillustration,c.f.Fig.letusconsideracloakwithinnerandouterbound-ariesexpressedas=1+0.1sin0.15cos+0.2sin+0.1cos=20.1cos0.15cos+0.3sin+0.2cosItisinterestingtoinspectthecloakparameterval-uesatitsnonreßectingouterboundary.Thiscanbedonethroughtheanalysisoftheentriesoftheinverseofthemetrictensorinthepolarbasis.WeÞrstnoticeinFig.thattheoff-diagonaltermsaregenerallynonzero,unlikethecir-cularcasewhenisdiagonal.Here,isdiagonalonlyforsixvaluesoftheanglecorrespondingtosixpointsoftheouterboundarywherethetransformedcoordinatesystemislocallyorthogonal.Thisdemon-stratesthatthepreviouscriteriaforanonreßecting,namely,(circularcase[])and(eccentricellipticcase[]),canbefurtherrelaxed.Ingeneralweob-servethat11,whichreßectstherotationofthetensorwithrespecttoitseigenbasis.In-deed,theprincipalrefractiveindicesofthecloakre-mainpositive:AlongweÞnd 0whereasalong TT Wefurthernotethatalsovarieswith,unlikeforcircular,elliptic,andeccentricellipticcloaks[.Last,01and1.5,inagreementwiththefactthatthecloakoughttoexhibitastrongazi-muthalanisotropyforthewavetobebentaroundit.Letusnowconsiderawiresourceantennacen-teredatpoint2.5,2.Itradiatesinavacuumwithwavelength=1(alllengthsaregiveninarbi-traryunits,micrometersfornearinfrared),anditcarriesacurrent.InFigs.,weshowthewavepatternoftheelectricÞeldscatteredbyaF-shapedlossyobstacleofpermittivitywhenitissurroundedbythecylindricalcloakde-scribedabove.Weobservesomeresidualinterfer-encesinFig..Foracomparablemesh,interferencesarelessapparentinthecircularandellipticcasescases8,12];thissuggeststheexperimentalrealizationofanarbitrarilyshapedcloakwouldbefairlychalleng-Last,itisinterestingtocomparethesenumericalresultscarriedoutintheintensenear-Þeldlimitwiththoseobtainedfromananalyticalapproach.Forthis,wecomputetheGreenÕsfunctionofHelmholtzÕsequa-tioninthetransformedcoordinates.TheelectricÞeldisthereforegivenbygivenbytheinversemapofthemapdeÞnedbyEq..WeusedthesoftwareMathematicatoproducethecorrespondingraypic-ture(seeFig.).ComparingthisÒidealÓcloakingwiththatofFigs.,wenoticethattheelectro-magneticeigenÞeldpenetratesthecloakdeeperthanintheraymodel.Thesingularbehaviorofontheinnerboundaryofthecloakmightbeincriminated.InthisLetter,weanalyzedtheelectromagneticre-sponseofanarbitrarilyshapedcloakinpresenceofacloselylocatedantenna.Wecomparedthefull-wavepicturewithananalyticcomputationoftheGreenÕsfunctioninthetransformedcoordinates.Wenoticedthestrongsimilaritiesinthenumericalresults,whichsuggeststhatthiscloakingmechanismde-pendslittleuponwhetherweareintherayopticsorresonanceregime.Ourdesigncanbeextendedstraightforwardlytothree-dimensionalcloaksofar-bitraryshape.Forthis,consideratransformedradialwheretheangleplacesthelongitudinalcoordinateinEq.ThendescribetheinnerandouterboundariesofthecloakusingtheFourierThisworkhasbeenperformedintheframeworkofthePOEMprojectANR-06-NANO-008.1.J.B.Pendry,Phys.Rev.Lett.,3966(2000).2.G.W.Milton,N.A.P.Nicorovici,R.C.McPhedran,andV.A.Podolskiy,Proc.R.Soc.LondonSer.A,39993.S.Guenneau,B.Gralak,andJ.B.Pendry,Opt.Lett.,1204(2005).4.J.B.Pendry,D.Schurig,andD.R.Smith,Science1780(2006).5.U.Leonhardt,Science,1777(2006).6.A.AluandN.Engheta,Phys.Rev.E,016623(2005).7.G.W.MiltonandN.A.Nicorovici,Proc.R.Soc.LondonSer.A,3027(2006).8.F.Zolla,S.Guenneau,A.Nicolet,andJ.B.Pendry,Opt.Lett.,1069(2007).9.D.Schurig,J.J.Mock,B.J.Justice,S.A.Cummer,J.B.Pendry,A.F.Starr,andD.R.Smith,Science977(2006).10.W.Cai,U.K.Chettiar,A.V.Kildiev,andV.M.Shalaev,Nat.Photonics,224(2007).11.D.KwonandD.H.Werner,Appl.Phys.Lett.013501(2008).12.A.Nicolet,F.Zolla,andS.Guenneau,IEEETrans.,1150(2008).13.M.Rham,D.Schurig,D.A.Roberts,S.A.Cummer,D.R.Smith,andJ.B.Pendry,PhotonicsNanostruct.Fundam.Appl.,87(2008).14.A.Nicolet,J.F.Remacle,B.Meys,A.Genon,andW.Legros,Appl.Phys.,6036(1994).15.P.Dular,C.Geuzaine,F.Henrotte,andW.Legros,IEEETrans.Magn.,3395(1998)(seealsohttp://OPTICSLETTERS/Vol.33,No.14/July15,2008